Inverse Haversine

Author

Eric W. Weisstein
August 24, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/InverseHaversine.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/InverseHaversine.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Notation

InverseHaversine[z]//TraditionalForm

Definition

InverseHaversine[z]//FunctionExpand//TraditionalForm

Plot

<<Utilities`Plot`

Real Axis

Plot[InverseHaversine[x],{x,0,1},AxesLabel->{x,InverseHaversine[x]}]

Complex Plane

ComplexContourPlot[InverseHaversine[z],{z,-5,5}]

ComplexPlot3D[InverseHaversine[z],{z,-5,5}]

Derivative and Integral

Derivative

D[InverseHaversine[z],z]//TraditionalForm

D[InverseHaversine[z],z]

Integrate[%,z]

FullSimplify[%]

Integral

Integrate[InverseHaversine[z],z]//TraditionalForm

Integrate[InverseHaversine[z],z]

D[%,z]

FullSimplify[%]

Series

Series[InverseHaversine[z],{z,0,10}]

Normal[Series[InverseHaversine[z],{z,0,10}]]/Sqrt[z]//Simplify

Through[{Numerator,Denominator}[CoefficientList[%,z]]]

Table[((Binomial[2n,n])/((4^n)(2n+1)))(x^n),{n,0,10}]

FullSimplify[((Binomial[2n,n])/((4^n)(2n+1))),n∈Integers&&n>0]

SeriesCoefficient[InverseHaversine[z],{z,0,n}]

Inverse

Haversine[InverseHaversine[z]]

Haversine[InverseHaversine[z]]//FullSimplify

InverseHaversine[Haversine[z]]//FullSimplify

Sums

(∑_(n=0)^∞)(((Binomial[2 n, n]) (x^(n+(1/2))))/((2^(2 n-1)) (2 n+1)))==(hav^(-1))(x)

Table[((Binomial[2n,n])/((2^(2n-1))(2n+1)))(x^(n+1/2)),{n,0,4}]

Sum[((Binomial[2n,n])/((2^(2n-1))(2n+1)))(x^(n+1/2)),{n,0,∞}]

FullSimplify[%-InverseHaversine[x]]